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Learning knot invariants across dimensions
by Jessica Craven, Mark Hughes, Vishnu Jejjala, Arjun Kar
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Submission summary
| Authors (as registered SciPost users): | Jessica Craven · Vishnu Jejjala |
| Submission information | |
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| Preprint Link: | scipost_202201_00036v1 (pdf) |
| Date submitted: | 2022-01-26 15:54 |
| Submitted by: | Craven, Jessica |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
We use deep neural networks to machine learn correlations between knot invariants in various dimensions. The three-dimensional invariant of interest is the Jones polynomial J(q), and the four-dimensional invariants are the Khovanov polynomial Kh(q,t), smooth slice genus g, and Rasmussen's s-invariant. We find that a two-layer feed-forward neural network can predict s from Kh(q,−q−4) with greater than 99% accuracy. A theoretical explanation for this performance exists in knot theory via the now disproven knight move conjecture, which is obeyed by all knots in our dataset. More surprisingly, we find similar performance for the prediction of s from Kh(q,−q−2), which suggests a novel relationship between the Khovanov and Lee homology theories of a knot. The network predicts g from Kh(q,t) with similarly high accuracy, and we discuss the extent to which the machine is learning s as opposed to g, since there is a general inequality |s|≤2g. The Jones polynomial, as a three-dimensional invariant, is not obviously related to s or g, but the network achieves greater than 95% accuracy in predicting either from J(q). Moreover, similar accuracy can be achieved by evaluating J(q) at roots of unity. This suggests a relationship with SU(2) Chern-Simons theory, and we review the gauge theory construction of Khovanov homology which may be relevant for explaining the network's performance.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2022-7-29 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202201_00036v1, delivered 2022-07-29, doi: 10.21468/SciPost.Report.5468
Report
In this paper the authors use machine learning techniques to find relations between various knot invariants: Khovanov and Jones polynomials, s-invariant, and slice genus. It is shown that a relatively simple neural network can very well predict s-invariant and slice genus from both Khovanov and Jones polynomials. The authors argue that successful predictions based on the Khovanov polynomial can be expected in view of 4-dimensional interpretation of the objects involved (Khovanov polynomial, s-invariant and slice genus), while the prediction based on the Jones polynomial is harder to explain in view of its 3-dimensional (rather than 4-dimensional) origin. The authors briefly speculate on interpretations of these results in terms of gauge theories.
The results of the paper are interesting, although not groundbreaking. It is not surprising that recently developed powerful techniques in machine learning can be applied to knots and links, for which a lot of various invariants can be computed and have been tabulated. Nonetheless, it is interesting to combine machine learning techniques with knot theory, and developing this research direction could presumably lead to some real breakthrough, as it happened in other fields. In addition, the first part of the paper is a nice summary of various results from knot theory, which should be useful for readers who are not experts in this area, but are interested in developing this research direction. For these reasons I recommend this paper for publication.
Report #1 by Anonymous (Referee 1) on 2022-7-8 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202201_00036v1, delivered 2022-07-08, doi: 10.21468/SciPost.Report.5367
Report
Dear editor,
This is a nice paper and I would recommend publication with minor revision.
Here are some comments:
Introduction is well written and includes good motivation as to why it is worthwhile studying the relationships between knot invariants in terms of working towards a dimension independent theory of knots.
Section 2 introducing Khovanov homology is clear and given at an appropriate level of detail.
W on page 5 is not defined. Is W in the definition of W{k} supposed to be V?
Section 3 describes the physical interpretation of the knot invariants in terms of gauge theory. This
section is well written and connection of the results to this is made in the discussion section.
In section 3.5 the s# invariant is introduced which is not equal to the s invariant. Could you use this work to deduce some relationship between these two? For instance perform multi-task learning like in Section 4.5.1.
Results are presented well in Section 4 and their interpretation discussed critically.
Using the same y-axis range in the plots in Figure 3 would make it easier to compare the two and again for plots in Figure 4.
In section 4.5.1 please add a short explanation to make it clear how the results of differing the weights for the two invariants tells us what the network is learning.
one typo in the first paragraph of section 2.1 on line 4 where they repeated ‘for for’.
Authors are commended for including their code script in the appendix. A link to a GitHub repo would also be helpful for readers.
Author: Jessica Craven on 2022-10-26 [id 2951]
(in reply to Report 1 on 2022-07-08)We are grateful for the suggested improvements to the text. We have included a list of changes along with our resubmission.
The suggestion of learning $s^\sharp$ is excellent, but we do not have sufficiently many examples of knots with different values of $s$ and $s^\sharp$ to perform the experiment. While the Rasmussen $s$-invariant is often readily computable, the Kronheimer and Mrowka $s^\sharp$-invariant is not. Indeed, the argument that these differ even for the trefoil is subtle and was originally missed in the mathematics literature.